QuestionSolve the equation $\frac{1}{2}|x+8|=|4 x-1|$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value function, $|x|$ , is defined as $x$ if $x \geq0$ and $-x$ if $x <0$ . . The equation $\frac{1}{}|x+8|=|4x-1|$ can be split into two separate equations based on the definition of the absolute value function.

STEP 2

We split the given equation into two separate equations based on the definition of the absolute value function. $\frac{1}{2}(x+8) =4x-1 \quad \text{if} \quad x+8 \geq0$ $\frac{1}{2}(-x-8) =4x-1 \quad \text{if} \quad x+8 <0$

STEP 3

We solve the first equation $\frac{1}{2}(x+8) =x-1$ for $x$ .
$x+8 =8x-2$

STEP 4

Rearrange the equation to isolate $x$ .
$8x - x =8 +2$

STEP 5

implify the equation to find the value of $x$ .
$7x =10$ $x = \frac{10}{7}$

STEP 6

We check if the solution $x = \frac{10}{}$ satisfies the condition $x+8 \geq0$ .
$\frac{10}{} +8 \geq0$

STEP 7

The condition is satisfied, so $x = \frac{10}{7}$ is a valid solution.

STEP 8

We solve the second equation $\frac{1}{2}(-x-8) =4x-1$ for $x$ .
$-x-8 =8x-2$

STEP 9

Rearrange the equation to isolate $x$ .
$8x + x =8 +2$

STEP 10

implify the equation to find the value of $x$ .
$9x =10$ $x = \frac{10}{9}$

STEP 11

We check if the solution $x = \frac{10}{9}$ satisfies the condition $x+8 <0$ .
$\frac{10}{9} +8 <0$

STEP 12

The condition is not satisfied, so $x = \frac{10}{9}$ is not a valid solution.
The solution to the equation $\frac{}{2}|x+8|=|4x-|$ is $x = \frac{10}{7}$ .

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